Notes on “Mathemata” in Heidegger’s “Modern Science, Metaphysics, and Mathematics”

1. Heidegger cites Kant as giving us a “fundamental feature of modern science:”

…modern science is mathematical. From Kant comes the oft-quoted but still little understood sentence, “However, I maintain that in any particular doctrine of nature only so much genuine science can be found as there is mathematics to be found in it” (Preface to Metaphysical Beginning Principles of Natural Science). (Heidegger 273)

One has to distinguish between “mathematics” and “mathematical:” “mathematics itself is only a particular formation of the mathematical” (273). The Greek ta mathemata, a plural noun, designates “what can be learned and thus, at the same time, what can be taught” (274). That learning and teaching are of a capacity to understand is of the utmost importance:

The mathemata are the things insofar as we take cognizance of them as what we already know them to be in advance, the body as the bodily, the plant-like of the plant, the animal-like of the animal, the thingness of the thing, and so on. This genuine learning is therefore an extremely peculiar taking, a taking where one who takes only takes what one basically already has…. If the student only takes over something that is offered he does not learn. He comes to learn only when he experiences what he takes as something he himself really already has (275).

Mathemata sound an awful lot like the Platonic forms: true learning is a reaching into the soul, a recollecting of what was forgotten when we came into this earthly life. Now do need to clarify, before I get more in-depth about other topics, how the mathemata actually relate to something we are familiar with:

Numbers are the most familiar form of the mathematical because, in our usual dealing with things, when we calculate or count, numbers are the closest to that which we recognize in things without deriving it from them. (277)

2. I’m not entirely clear on whether Heidegger is entirely approving of Platonic philosophy in the lecture as a whole:

Learning is a kind of grasping and appropriating. But not every taking is a learning. We can take a thing, for instance, a rock, take it with us and put it in a rock collection. We can do the same with plants. It says in our cookbooks that one “takes,” i.e. uses (275).

The issue is whether categorization through use of logos is the same thing as literally taking things in the world apart for one’s own possession and use. Isn’t rock collection to some degree the same as categorizing types of rocks? Don’t we need to collect rocks to compare? Perhaps Platonic conceptualization is not entirely innocent of violence of a sort. If dialectic does result in a drawing forth from the soul, we note that it does not treat Socratic interlocutors very kindly.

Still, Heidegger in this section (“The Mathematical, Mathesis“) comments favorably on Socrates and Plato. He articulates nothing less than the core of the philosophical project:

The most difficult learning is coming to know actually and to the very foundations what we already know. Such learning, with which we are here solely concerned, demands dwelling continually on what appears to be nearest to us, for instance, on the question of what a thing is. We steadfastly ask the same question – which in terms of utility is obviously useless – of what a thing is, what tools are, what man is, what a work of art is, what the state and the world are. (276)

I think if we choose to attack Heidegger for not saying that “What is justice?” is the critical question, we’re nit-picking. Inasmuch I know something about philosophy itself, I do think it is the critical question, and not just for generating political philosophy. “Appears to be nearest to us” as the site of dwelling is not a neutral statement. Take careful note of “appears.” I suggest the question of justice is right beneath the surface here.

Heidegger comments on the life of Socrates to illustrate the above passage about “the most difficult learning.” The passage has especial relevance for me as the source is Xenophon. The unnamed scholar/Sophist is Hippias, noted for his flashy appearance and bragging about the money he made (see Greater Hippias):

In ancient times there was a famous Greek scholar who traveled everywhere lecturing. Such people were called Sophists. This famous Sophist, returning to Athens once from a lecture tour in Asia Minor, met Socrates on the street. It was Socrates’ habit to hang around on the street and talk with people, for example, with a cobbler about what a shoe is. Socrates had no other topic than what the things are. “Are you still standing there,” condescendingly asked the much-traveled Sophist of Socrates, “and still saying the same thing about the same thing?” “Yes,” answered Socrates, “that I am. But you who are so extremely smart, you never say the same thing about the same thing.” (276)

I do say that Socrates possessed the beings based on this story and a few other passages in the Memorabilia. But we need not go that far to understand the importance of mathemata. It seems he has mathemata, and they are probably apprehended natures. The relation between “being” and “nature:” “With Aristotle, however, this “force,” dynamis, the capacity for its motion, lies in the nature of the body itself” (285). Nature seems to be being in time, at the least (in Introduction to Metaphysics: nature is the thing’s striving toward being).

Is Hippias wrong to change his opinion? Not at all. Only: he was changing his opinion for money and thus moving from place to place. (We learn later in the essay that “nature” includes place. Bodies move toward the places they belong – 285). The Socratic investigation: ask someone with a natural talent for doing or making something what he does or makes. The inquiry’s conclusion – the inquiry itself – only appears to be at hand.

3. “Mathematical” can be the things we are capable of learning in the more or less Socratic way. But it is also “the manner of learning and the process itself…. [the] fundamental presupposition of the knowledge of things” (278-9). With this stated, Heidegger introduces Plato:

…Plato put over the entrance to his Academy the words: Ageometretos medeis eisito! “Let no one who has not grasped the mathematical enter here!” These words do not mean that one must be educated in only one subject – “geometry” – but that one must grasp that the fundamental condition for the proper possibility of knowing is knowledge of the fundamental presuppositions of all knowledge and the positions we take based on such knowledge. A knowledge which does not build its foundation knowledgeably, and thereby notes its limits, is not knowledge but mere opinion. The mathematical, in the original sense of learning what one already knows, is the fundamental presupposition of “academic” work. (278)

Again, I think Heidegger is basically correct. I might be more explicit about a tension between self-knowledge and knowledge. The Platonic and Xenophontic dialogues definitely work with the theme that love of wisdom is hubris. It isn’t even clear to me that self-knowledge exists, and I know this question can be advanced through reading Plato.

Is a potential Heideggerian criticism of Plato hiding here? It may be the case that “mathematical” already has overtones of a narrower vision that characterizes modern scientists (if not modern science). If I blog more on this essay, it will be very clear that “nature” is central to Aristotlean physics and that “mathematics” in modern philosophic thought regarding science is used to displace nature.


Heidegger, Martin. “Modern Science, Metaphysics, and Mathematics” in Basic Writings, ed. Krell. New York: HarperCollins, 2008. 271-305.


  1. Wow a lot of that was hard to understand but what i was able to take from it was that from our pre learned knowledge we are able to reason and compute to a certain degree. everything in this world is based on mathematics, down to the petals on a rose. Math is within everything.

  2. I have to say that I do believe that it is great deal of that has been hard to understand but things I could take from it had been that from your pre learned knowledge we are able to reason and compute to a certain degree.

  3. It is interesting to note how frequently Heidegger relies on the ancient Greek language to support his arguments. I wonder how Kant would respond to Heidegger’s discussion of Socrates here in relation to Kant’s original proposition.

  4. I am wondering about the relations between Mathemata and Mnemosyne—-naming recalling recounting accounting…

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